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Control charts are among the most effective management control tools, and they are as important as cost controls and material controls. Modern computer technology has made it easy to implement control charts in any type of process, because data collection and analysis can be performed on a microcomputer or a local area network terminal in realtime, online at the work center.

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To illustrate these ideas, we give a simpli ed example of a control chart. In manufacturing automobile engine piston rings, the inside diameter of the rings is a critical quality characteristic. The process mean inside ring diameter is 74 millimeters, and it is known that the standard deviation of ring diameter is 0.01 millimeters. A control chart for average ring diameter is shown in Fig. 16-3. Every hour a random sample of ve rings is taken, the average ring diameter of the sample (say x) is computed, and x is plotted on the chart. Because this control chart utilizes the sample mean X to monitor the process mean, it is usually called an X control chart. Note that all the points fall within the control limits, so the chart indicates that the process is in statistical control. Consider how the control limits were determined. The process average is 74 millimeters, and the process standard deviation is 0.01 millimeters. Now if samples of size n 5 are taken, the standard deviation of the sample average X is 1n 0.01 15 0.0045





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Pn,s = -Ten,s,

Therefore, if the process is in control with a mean diameter of 74 millimeters, by using the central limit theorem to assume that X is approximately normally distributed, we would expect approximately 100(1 )% of the sample mean diameters X to fall between 74 z 2(0.0045) and 74 z 2(0.0045). As discussed above, we customarily choose the

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where T is any constant, they acquire the form of Hamilton's equations for a set of uncoupled harmonic oscillators

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c1n,s = Pn,s Pn,s = -c2k~qn,s,

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as shown on the control chart. These are the 3-sigma control limits referred to above. Note that the use of 3-sigma limits implies that 0.0027; that is, the probability that the point plots outside the control limits when the process is in control is 0.0027. The width of the control limits is inversely related to the sample size n for a given multiple of sigma. Choosing the control limits is equivalent to setting up the critical region for testing the hypothesis H0: H1: 74 74

where 0.01 is known. Essentially, the control chart tests this hypothesis repeatedly at different points in time. In designing a control chart, we must specify both the sample size to use and the frequency of sampling. In general, larger samples will make it easier to detect small shifts in the process. When choosing the sample size, we must keep in mind the size of the shift that we are trying to detect. If we are interested in detecting a relatively large process shift, we use smaller sample sizes than those that would be employed if the shift of interest were relatively small. We must also determine the frequency of sampling. The most desirable situation from the point of view of detecting shifts would be to take large samples very frequently; however, this is usually not economically feasible. The general problem is one of allocating sampling effort. That is, either we take small samples at short intervals or larger samples at longer intervals. Current industry practice tends to favor smaller, more frequent samples, particularly in high-volume manufacturing processes or where a great many types of assignable causes can occur. Furthermore, as automatic sensing and measurement technology develops, it is becoming possible to greatly increase frequencies. Ultimately, every unit can be tested as it is manufactured. This capability will not eliminate the need for control charts because the test system will not prevent defects. The increased data will increase the effectiveness of process control and improve quality.

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